This 1-year study will enroll individuals with advanced blood diseases or cancer. Participants will receive an allogeneic umbilical cord blood stem cell transplant and will be closely monitored while in the hospital to determine when engraftment occurs and if GVHD develops. Study visits will occur at Months 1, 2, 3, 6, 9, and Year 1. At each visit, participants will have blood drawn for laboratory testing and for evaluation of immune system response. Quality of life questionnaires will also be completed at each visit.